Episode 3: Derivatives - The Mechanical Universe – Transcript

00:00foreign

01:05thank you differential calculus is a powerful mathematical tool for analyzing how things change the handles on this tool are a few simple rules for calculating derivatives somewhere around 600 BC someone discovered that if you want to make Pleasant sounding chords on a

01:32stringed instrument then the lengths of the strings ought to be in the ratio of simple numbers like one to two or two to three and so on these are called the Pythagorean harmonies the discovery was very important because it was the very first discovery of a connection between mathematics and the physical world unfortunately that connection was forgotten and had to

01:58be rediscovered slowly and painfully thousands of years later one person who certainly grasped it though was Galileo Galilei in any case I want to read to you something that Galileo wrote this book was published in Rome in 1623 and it's called yo sagittori which is usually translated to mean the

02:26assaya but I prefer to call it the experimentalist because I think that's more like what Galileo had in mind Galileo had this nasty habit of making his famous remarks in Italian but I'll translate it for you as I go along he said true knowledge is written in this enormous book which is continuously open before our

02:50eyes I speak of the universe but one can't understand that unless first one learns to understand the language and recognize the characters in which it's written it's written in the language of mathematics so to prepare to read the book of the universe we must first learn the symbols and the vocabulary of the language of

03:18mathematics it's a language of precision and poetry and even of music for a long time physicists have used the language of mathematics and since somewhere around 600 BC so have musicians as with just about all languages

03:44including music mathematics has its own vocabulary its own rules and symbols its precision and elegance it's poetry and its history one part of that history was Galileo Galilei who was something of a non-conformist it was a trait he inherited from his father Vincenzo who was an accomplished

04:11musician musically Vincenzo refused to be bound by traditional forms a position that would become the family trademark he wrote a book opposing the use of Pythagorean harmonies among his musical contemporaries he considered the ancient Greek gods too simple for the complex musical structures of the Italian Renaissance

04:40later like father like son Galileo considered the Greek standards of mathematics too simple to express his ideas created the science of kinematics a branch of mechanics dealing with Motion in the abstract and if any abstract idea is to be expressed properly it needs appropriate language

05:06the concepts and symbols that give an idea its meaning and value as advanced as it was Galileo's new science of motion was still rooted in the soil of the ancient Greek intellect and for anything entirely new to bloom in the fields of mathematics and science Scholars needed a language more

05:31sophisticated than the one that had been spoken since Archimedes and Euclid in other words after Galileo physics needed an advanced language fewer than 25 years after his death that very language would be discovered and in one form or another spoken ever after

05:55it would come to be called differential calculus differential calculus is very powerful and as with any language it derives its power from the idea behind it the derivative the derivative is to kinematics what the wheel is to travel

06:19the simple yet very effective means for getting from one place to another and to get just the right perspective of what else a derivative is nothing works better than a little exercise to begin with a derivative doesn't apply only to a body in horizontal motion nor for that matter only to a body in vertical motion

06:47up down any which way the derivative is the rate of change of any function at any exact point or instant as illustrated by Galileo's law of falling bodies speed is the derivative of distance but it's more than that a derivative can represent the rate of change of anything

07:13for example the population density of dolphins in relation to increasing or decreasing water temperature or the rate of change in the volume of a balloon versus its surface area or even the rate of change in the price of a pizza with respect to its size clearly then the concept of the derivative goes far and wide

07:41but the mechanical process of the derivative differential calculus needs a practical approach or the concept goes nowhere fast eventually without the rules of differentiation the simple concept of the derivative becomes an uphill struggle in the long run it helps to pick up a few more definitions along the way

08:07so before it's too late to turn back consider the factor of steepness on the incline the steepness is the ratio of the change in elevation did the change in horizontal distance this ratio a number is called the slope for example suppose the elevation of an incline

08:33increases 15 meters every 100 meters the rider moves upward 15. horizontally 100. the slope is 0.15 a hill with a slope of 0.3 is twice as steep as one with a slope of 0.15 bigger the slope the steeper the hill

09:01when the slope is large it's no small feat to get to the top when the slope is next to nothing near zero it's easy going and when the slope is negative it's downhill all the way although mathematics can be easy going it does have its Peaks and valleys and it always has

09:26nobody knows who first asked the best way to get from here to there but the answer in algebraic terms is first offered by a French mathematician named fromat in 1629 he came up with the idea of finding the tangent line to an arbitrary point on a curve

09:50in 1638 Vermont shared his Discovery with his countrymen Rene Descartes who had his own method for finding tangents to algebraic curves many of these mathematical ideas simply those are fromat were further developed by Wilhelm leibniz and Isaac Newton

10:16into a general and systematic method of mathematical analysis differential calculus setting history aside at least for the present some timely questions remain for example on a smoothly changing curve there's a constantly changing slope

10:43how then in today's language is a slope calculated at any given point to determine the slope at a particular Point here for example simply take another point on the hill it doesn't matter where now connect the two points with a

11:07straight line that line is called a chord and its slope depends on the location of the second point the first and second points are reasonably close the chord is a reasonably good approximation of the bike's path now move the second Point closer to the first

11:32move it even closer the slope is a number and as the points get closer together the number gets closer to a certain value it's reasonable to call that number the slope of the hill at that point the line with that slope through the point is called the tangent line and the tangent line is just what the chord turns into as the points get closer together

11:59and the slope of the tangent line at that point is the slope of the Hill instantaneous speed can be calculated along the same lines yes oh no Galileo's law of falling bodies applied here to a rather reluctant one is more than an experiment in tariff

12:24its differential calculus to the rescue change in distance is divided by change in time the ratio is the average speed during a given time interval when that time streaks to zero the limiting value of the average speed is the instantaneous speed

12:48change in elevation is divided by change in horizontal distance the result is the slope of a chord joining two points when the horizontal distance shrinks to zero value of the slope of the chord is the slope at that point differentiation the goals and calculations differ but not the essential concept nor the

13:17process speed is a derivative of distance with respect to time slope is the derivative of elevation with respect to horizontal distance in any case a derivative is what happens to a quotient a ratio of two numbers has both top and